Every topological group G naturally embeds in the Banach algebra LUC(G)*. The topological centre of LUC(G)* is defined to be the set of its elements for which the left multiplication is w*–w*-continuous. Although the definition demands continuity on the whole algebra, for a large class of topological groups it is sufficient to test the continuity of the left multiplication at just one suitably chosen point; in other words, the algebra has a one-point DTC (Determining Topological Centre) set. More generally, the same result holds for many subsemigroups of LUC(G)*. In particular, for G in the same large class, the uniform compactification (the greatest ambit) of G has a one-point DTC set. These results, which generalize those previously known for locally compact groups, are from joint work with Stefano Ferri and Matthias Neufang.